Riemann's removable-singularity theorem

You meet a function holomorphic everywhere on a punctured disk except possibly at the centre, and you want to know whether the centre is a genuine singularity or just a fillable hole. Riemann's theorem gives a startlingly simple test: just check whether the function stays bounded near the point. If it does, the singularity was never real — it can be removed.

Precisely: let f be holomorphic on the punctured disk 0 < |z - z_0| < r. If f is bounded there (its absolute value stays below some fixed number M as z approaches z_0), then z_0 is a removable singularity — f extends to a function holomorphic on the whole disk |z - z_0| < r. You do not even have to know the limit in advance; mere boundedness forces the principal part of the Laurent series to vanish. A quick way to see why: a_{-n} = (1/(2 pi i)) times the integral of f(z)(z - z_0)^(n-1) dz over a small circle of radius rho; the ML-inequality bounds this by M times rho^n, which goes to 0 as rho shrinks for n greater than or equal to 1, so every negative coefficient is forced to be zero.

This theorem is the reason boundedness is the litmus test for the mildest singularity. It is also surprisingly powerful as a tool: it lets you ignore points where you only know a function is bounded, knowing afterward it was holomorphic there all along. It contrasts sharply with poles (|f| goes to infinity) and essential singularities (|f| has no limit at all), so the three types are cleanly separated by their behaviour: bounded, blows up to infinity, or wild.